Found problems: 1513
Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(f(x)+x+f(y)f(z))=f(x+zf(x)+zf(y))-xf(z-1)\]for all $x,y,z\in\mathbb{R}$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$
Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$.
Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ?
(I. Gorodnin)
Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\]
holds for all $x,y \in \mathbb{R}$.
Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds:
$$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that
$$|xf(y)-yf(x)|$$
is a perfect square for every $x,y \in \mathbb{N}$
Find all $f:\mathbb{R}\to\mathbb{R}$ functions such that
$$f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)$$
for all real numbers $x,y$
Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold
$$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$
$$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$
[i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]
Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\] for all reals $x$, $y$.
[i](Proposed by Lim Yun Zhe)[/i]
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that for all reals $ x, y $,$$ f(x^2+f(x+y))=y+xf(x+1) $$
Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration:
$$ \int_{0}^{\infty} f(x) \,dx $$
Determine all real functions $f(x)$ that are defined and continuous on the interval $(-1, 1)$ and that satisfy the functional equation
\[f(x+y)=\frac{f(x)+f(y)}{1-f(x) f(y)} \qquad (x, y, x + y \in (-1, 1)).\]
Consider all functions $f:\mathbb{N}\to\mathbb{N}$ satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.
find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st
$f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$
related:
https://artofproblemsolving.com/community/c6h381298
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$
Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$:
\[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]
Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$