Found problems: 2349
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$.
(Walther Janous)
We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that
\[
g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}.
\]
a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$
b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$
Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\] for all real numbers $x,y,z$.
Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.
Find all functions $f:(0,\infty)\to(0,\infty)$ such that
$$f(f(f(x)))+4f(f(x))+f(x)=6x.$$
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$.
Find all functions $f :\Bbb{ R}\to \Bbb{Z}$ such that $$\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),$$ for all $x, y \in \Bbb{R}.$
Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
(a) $f(1)=1$
(b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$.
(c) $f(n)$ divides $f(m)$ if $m>n$.
Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.)
[i]Proposed by Sean Li[/i]
Find all functions $f : \mathbb R \to \mathbb R$ such that for all reals $x, y, z$ it holds that
\[f(x + f(y + z)) + f(f(x + y) + z) = 2y.\]
Find all functions $f : \mathbb R \to \mathbb R$ such that
\[f(x)+f\left(1-\frac{1}{x}\right)=x,\]
holds for all real $x$.
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$
[i]Proposed by Mihai Baluna, Romania[/i]
Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and
\[2f(m)f(n)-f(n-m)-1\]
is a perfect square for all integer $m,n.$
Let $S$ be the set of all real numbers greater than or equal to $1$.
Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.
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]
Consider a function $f:\mathbb N\rightarrow \mathbb N$ such that for any two positive integers $x,y$, the equation $f(xf(y))=yf(x)$ holds. Find the smallest possible value of $f(2007)$.
Find the continuous function $f(x)$ such that the following equation holds for any real number $x$.
\[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\]
[i]1977 Keio University entrance exam/Medicine[/i]
For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by
\[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\]
Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.
Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]
Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying
$f(xy+c)=f(x)+f(y)$, for all $x,y \in S$
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that
\[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \]
[i]Călin Popescu[/i]
Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.
find all function $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(x^3-xf(y)^2)=xf(x+y)f(x-y)\]
holds for all real number $x$, $y$.
[i]Proposed by chengbilly[/i]
Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition:
for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.