Found problems: 1513
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(f(x-y)-yf(x))=xf(y).$$
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$,
$$f(xy+f(x))=f(xf(y))+x$$
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied.
i. $f(n+1) > f(n)$
ii. $f(f(n)) = 2n+2$
Find the value of $f(2013)$.
(Here, $\mathbb{N}$ is the set of all positive integers.)
Determine all functions $f:\mathbb{R}_+\to\mathbb{R}_+$ which satisfy \[f\left(\frac{y}{f(x)}\right)+x=f(xy)+f(f(x)),\]for any positive real numbers $x$ and $y$.
[i]Proposed by Pavel Ciurea[/i]
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 $n$.
Find all functions $f : \mathbb R \to \mathbb R$ such that
\[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
Find all functions $f : \mathbb R \to \mathbb R$ for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\]
for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
Let $H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}$. Describe all functions $f : H \to H$ for which
a) $x = f(x) - f(f(x))$ holds for every $x \in H$.
b) $x = f(x) + f(f(x)) - f(f(f(x)))$ holds for every $x \in H$.
c) $x = f(x) + 2f(f(x)) - 3f(f(f(x)))$ holds for every $x \in H$.
PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2
Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$ and
a) $f(1-f(1))\ne 0$
b) $ f(1-f(1))= 0$
S. Kuzmich, I.Voronovich
Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$.
I.Voronovich
Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.
Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]
Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$,
$$f(xf(y)+y)=yf(x)+f(y).$$
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]