Found problems: 2349
Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$.
[i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
Let $f$ be a map of the plane into itself with the property that if $d(A,B)=1$, then $d(f(A),f(B))=1$, where $d(X,Y)$ denotes the distance between points $X$ and $Y$. Prove that for any positive integer $n$, $d(A,B)=n$ implies $d(f(A),f(B))=n$.
Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$.
[i]Dinu Teodorescu[/i]
For some positive integer $n,$ Elmo writes down the equation
\[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\]
Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation
\[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\]
Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$?
[i]Srinivas Arun[/i]
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satidfy :
$$2f(x+y+xy)= a f(x)+ bf(y)+f(xy)$$ for any $x,y \in\mathbb{R}$ όπου $a,b\in\mathbb{R}$ with $a^2-a\ne b^2-b$
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
Set $\mathbb Q_1=\{x\in\mathbb Q\mid x\ge1\}$. Suppose that a function $f:\mathbb Q_1\to\mathbb R$ satisfies the inequality $\left|f(x+y)-f(x)-f(y)\right|<\epsilon$ for all $x,y\in\mathbb Q_1$, where $\epsilon>0$ is given. Prove that there exists a real number $q$ such that
$$\left|\frac{f(x)}x-q\right|<2\epsilon\qquad\text{for all }x\in\mathbb Q_1.$$
Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
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.
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation
$$ xf(x/a)-f(a/x)=b, $$
where $ a\neq 0,b $ are two real numbers.
[i]Dan Popescu[/i]
Find all polynomials $P$ in two variables with real coefficients satisfying the identity
$$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.
Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality:
\[f(3x+2y)=f(x)f(y)\]
for all non-negative integers $x,y$.
Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that:
(i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$.
(ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$.
[i]Proposed by Yang Liu[/i]
[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b]
Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:
$\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$
A function $f : R -\{0\} \to R$ has the following properties:
(i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$,
(ii) $f$ takes the value $\frac12$ at least once.
Determine $f(-1)$.
Prove that $f$ is a periodic function
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]