Found problems: 1513
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $
Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $
Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.
A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.
Find all $a\in\mathbb R$ for which there exists a function $f\colon\mathbb R\rightarrow\mathbb R$, such that
(i) $f(f(x))=f(x)+x$, for all $x\in\mathbb R$,
(ii) $f(f(x)-x)=f(x)+ax$, for all $x\in\mathbb R$.
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$
$$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$.
[i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying
\[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]
for all integers $x$.
[i]Proposed by Ankan Bhattacharya[/i]
Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that:
$af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$
For all positive real $x$ and large enough $y$.
Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that:
$f(xy)+f(\frac{x}{y})=2f(x)+h(y)$
For all positive real $x$ and large enough $y$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that
\[
f(x + f(y) - 2y) + f(f(y)) = f(x)
\]
for all integers \( x \) and \( y \).
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that
$$f(f(x)+f(y))+cxy=f(x+y).$$
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions:
[list]
[*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$
[*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$
[/list]
Prove that $a_n=n$ for all positive integers $n$.
[i]Proposed by José Alejandro Reyes González[/i]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$
for all $x, y \in \mathbb{R}$.
[i]Proposed by Md. Fuad Al Alam[/i]
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
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 $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]
For all $x,y\in\mathbb R$.
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$\bullet$ $f(x)<2$ for all $x\in (0,1)$;
$\bullet$ for all real numbers $x,y$ we have:
$$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$
Proposed by Navid Safaei