Found problems: 1513
Let \( \mathbb{R} \) denote the set of real numbers. Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[
f(x^2) - y f(y) = f(x+y)(f(x) - y)
\]
for all real numbers \( x \) and \( y \).
Find all polynomials $P(x)$ with real coefficients that satisfy the relation
$$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying
\[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\]
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set
\[X_v=\{x\in\mathbb Z:f(x)=v\}\]
is finite and nonempty.
(a) Prove that there exists such a function $f$ for which there is an $f$-rare integer.
(b) Prove that no such function $f$ can have more than one $f$-rare integer.
[i]Netherlands[/i]
Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$
Find all functions $f:\mathbb{N}_0\to\mathbb{N}$ such that
[b]1)[/b] \(f(a)\) divides \(a\) for every \(a\in\mathbb{N}_0\), and
[b]2)[/b] for all \(a,b,k\in\mathbb{N}_0\) we have
\[
f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr).
\]
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 $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\] for all positive real numbers $x$ and $y$.
[i](Andrew Brahms, USA)[/i]
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that
$x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]
Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$
Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ with the property that $$f(x+f(y)+f(f(z)))=z+f(y)+f(f(x))$$ for all positive integers $x,y,z$.
Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$.
a) Prove that $f(n)$ is an integer for all $n$.
b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.
Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$ holds for all positive integers $ m,n $
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.)
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and
\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]
for all $x, y \in\mathbb{R}$.
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that
(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$
(ii) $ f(0)=1,$ and
(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
Prove that $f$ is unique, and express $f(x)$ in closed form.
Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$
Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.
Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds
\[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
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 \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that
$$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$
$$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$
Prove that for every natural number $n$ the following inequality holds:
$$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$
Let $N$ be the set of positive integers.
Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.