Found problems: 2349
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds:
$$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$
[i]Proposed by Anzo Teh Zhao Yang[/i]
Find all functions $f:\mathbb{R}\to \mathbb{R}$ so that for any reals $x,y$ the following holds:
\[f(x\cdot f(x+y))+f(f(y)\cdot f(x+y))=(x+y)^2\]
Find all functions $f: \mathbb{R}\to{R}$ such that for any $a<b$, $f([a,b])$ is an interval of length $b-a$
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1)$$
for any $x,y$ integers.
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
Find all functions $f:\mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, such that $$xf(1+xf(y))=f(f(x)+f(y))$$ for all positive reals $x, y$.
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$?
[b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of
\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$
[b]iii.)[/b] The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$
Find all functions $f:\mathbb Z^+\to\mathbb R^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to\infty}\dfrac{f(n+1)}{f(n)}=1$.
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
I.Voronovich
Let $\mathbb{S}$ be the set of all functions $f:\mathbb{Z}\rightarrow \mathbb{R}$. Now, consider the function $g:\mathbb{S} \rightarrow \mathbb{S} ,g(f(x)) = f(x + 1)-f(x)$. Now, we call a function $f \in \mathbb{S}$ good if $g^n(f(x))=0$ for some natural $n$.
Prove that if $s \not = t \in S$ are good functions then $s(m)-t(m)$ is 0 for only finitely many $m \in \mathbb{Z}$.
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that
\[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\]
Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and
\[f(x+y)=f(x)+f(y)\]
And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$
Find all functions $f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that : $f(x)f(y)f(z)=9f(z+xyf(z))$, where $x$, $y$, $z$, are three positive real numbers.
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that
[b]a)[/b] $f(p)=1$ for every prime $p$.
[b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$
Find the least number $n \ge 2021$ such that $f(n)=n$
Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions:
1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied.
2) For every pair of real numbers $x$ and $y$,
\[ f(xf(y))+yf(x)=xf(y)+f(xy)\]
is satisfied.
Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$.
Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
[i]Fajar Yuliawan, Indonesia[/i]
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.