Found problems: 2349
Let $\mathbb{Z}$ be the set of integers. Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that:
$$2023f(f(x))+2022x^2=2022f(x)+2023[f(x)]^2+1$$
for each integer $x$.
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 real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that
\[f(x\plus{}y)\plus{}f(y\plus{}z)\plus{}f(z\plus{}x)\ge 3f(x\plus{}2y\plus{}3z)\]
for all $x, y, z \in \mathbb R$.
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ such that $ f (x + y) \le f (xy) $ for every pair of real $ x $, $ y$.
Find all functions $f:Q\rightarrow Q$ such that \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \] for all $x,y,z,t\in Q.$
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 $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:
[list]
[*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
[*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.
[/list]
[i]Proposed by Sutanay Bhattacharya[/i]
Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$.
[i]Proposed by Ryan Alweiss[/i]
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$
Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]
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]
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any reals $x, y,$ $$f(x + y)f(x - y) = (f(x))^2 - (f(y))^2$$. Additionally, suppose that $f(x + 2 \pi) = f(x)$ and that there does not exist a positive real $a < 2 \pi$ such that $f(x + a) = f(x)$ for all reals $x$. Show that for all reals $x$, $$ |f(\frac{\pi}{2})| \geq f(x)$$.
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]
The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that
\[
f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
\]
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 $f:\mathbb{N}_0 \to \mathbb{Z}$ such that $$f(k)-f(l) \mid k^2-l^2$$ for all integers $k, l \geq 0$.
Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.