Found problems: 313
Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.
Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$
(Vadym Koval)
Find all functions $f : (0, +\infty) \to (0, +\infty)$ which are increasing on $[1, +\infty)$ and for all positive reals $a, b, c$ they fulfill the following relation $f(ab)f(bc)f(ca)=f(a^2b^2c^2)+f(a^2)+f(b^2)+f(c^2)$.
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied:
i) $2f (x) + 2f (y) \le f (x + y)$
ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.
It is known about the function $f : R \to R$ that
$\bullet$ $f(x) > f(y)$ for all real $x > y$
$\bullet$ $f(x) > x$ for all real $x$
$\bullet$ $f(2x - f (x)) = x$ for all real $x$.
Prove that $f(x) = x + f(0)$ for all real numbers $x$.
Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$
for all $x, y$ positive reals.
Let $ f (n) $ be a function that fulfills the following properties:
$\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $.
$\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $.
$\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \right) = | f (b) -f (a) | $.
Find $ \displaystyle f (2009 ^ {2009 ^ {2009}}) $ and justify your answer.
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 $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?
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 $x,y > 0$.
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.
Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$.
Nguyễn Duy Thái Sơn
Is there a function $f(x)$, which satisfies both of the following conditions:
a) if $x \ne y$, then $f(x)\ne f(y)$
b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?
Find all functions $f: R \to R$ that satisfy
$$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$
Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy:
(a) for all $n,f(n)$ is a positive integer.
(b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$.
(c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .