Found problems: 2349
A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$. Given that $f(2548) = -2548$, find the value of $f(2)$.
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$
for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$
[i]Proposed by Liam Baker, South Africa[/i]
Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and:
$$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$
Where $\overline{a}$ is the complex conjugate of $a$.
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that
\[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\]
for all $x,y \in \mathbb{R} $.
Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.
Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$.
Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that:
\[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(f(x - y)) = f(x)f(y) + f(x) - f(y) - xy.\]
Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]
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 $x,y \in \mathbb{R}$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y)
$$
for all real numbers $x$ and $y$.
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
Find all functions $f: R \to R$ such that, for any $x, y \in R$:
$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$
Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.
Let $ f : [0, 1] \to\ R $ be a function such that :-
$1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ .
$2.)$ $f(1) = 1$ .
$3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ .
Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\] for all $x,y$.
Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$
and $g(1)=-8$
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$
[i]Author: Hojoo Lee, South Korea[/i]