Found problems: 2349
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]
Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$ for all real numbers $x$ and $y.$
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.
For all real numbers $x$, we denote by $\lfloor x \rfloor$ the largest integer that does not exceed $x$. Find all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy the equality
\[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\]
for all real numbers $x$ and $y$.
[i]Navneel Singhal, India[/i]
Let $\mathbb{R}^{+}$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^{+} \to \mathbb{R}^{+}$ satisfying for all $x, y \in \mathbb{R}^{+}$ the equality
\[f(xf(y))=f(xy)+x\]
For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$
Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$
\[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.
Determine all functions $f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}$ such that
$$f(x^2yf(x))+f(1)=x^2f(x)+f(y)$$
holds for all nonzero real numbers $x$ and $y$.
Find all $ f: \mathbb{R} \to \mathbb{R}$, such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all real $x,y$.
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$
for all $z\in \mathbb{C}$.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold:
$$f(xf(y)-yf(x))=f(xy)-xy$$
[i]Proposed by Dusan Djukic[/i]
Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true :
$$f(xf(x)+yf(y)) = xy$$
Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that
\[xy(f(x)-f(y))|x-f(f(y))\]
holds for all positive rationals $x$, $y$
(we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$)
[i]Proposed by supercarry & windleaf1A[/i]
Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that:
$f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$
by Greece, Athanasios Kontogeorgis (aka socrates)
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.
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]
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
We denote the set of nonzero integers and the set of non-negative integers with $\mathbb Z^*$ and $\mathbb N_0$, respectively. Find all functions $f:\mathbb Z^* \to \mathbb N_0$ such that:
$a)$ $f(a+b)\geq min(f(a), f(b))$ for all $a,b$ in $\mathbb Z^*$ for which $a+b$ is in $\mathbb Z^*$.
$b)$ $f(ab)=f(a)+f(b)$ for all $a,b$ in $\mathbb Z^*$.
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following.
$f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$
This questions comprises two independent parts.
(i) Let \(g:\mathbb{R}\to\mathbb{R}\) be continuous and such that \(g(0)=0\) and \(g(x)g(-x)>0\) for any \(x > 0\). Find all solutions \(f : \mathbb{R}\to\mathbb{R}\) to the functional equation
\[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\]
(ii) Find all continuously differentiable functions \(\phi : [a, \infty) \to \mathbb{R}\), where \(a > 0\), that satisfies the equation
\[(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.\]
Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $
Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]
[i]Jaedon Whyte[/i]