Found problems: 2349
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 $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $2f(x^2+y^2)=(x+f(y))^2+f(x-f(y))^2$ for all $x,y\in\mathbb{R}$.
Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$
Does there exist a twice differentiable function $f:\mathbb R\to\mathbb R$ such that $f'(x)=f(x+1)-f(x)$ for all $x$ and $f''(0)\ne0$? Justify your answer.
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 $n$.
Assume that the system of differential equations $y'=-z^3$, $z'=y^3$ with the initial conditions $y(0)=1$, $z(0)=0$ has a unique solution $y=f(x)$, $z=g(x)$ defined for real $x$. Prove that there exists a positive constant $L$ such that for all real $x$,
$$f(x+L)=f(x),\enspace g(x+L)=g(x).$$
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.
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]
Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]
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 \plus{} y)(f(x) \minus{} f(y)) \equal{} (x \minus{}y)f(x \plus{} y)$ for all $ x, y\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 $f(m) = 2m$.
Defined all $f : \mathbb{R} \to \mathbb{R} $ that satisfied equation $$f(x)f(y)f(x-y)=x^2f(y)-y^2f(x)$$ for all $x,y \in \mathbb{R}$
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 $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.
Let F(x) be a known distribution function, the random variables $\eta_1 , \eta_2 ...$ be independent of the common distribution function $F( x - \vartheta)$, where $\vartheta$ is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of $\varepsilon> 0$ there exist a Lebesgue measure $\varepsilon$ Borel set E ("confidence set") and a Borel-measurable function $t_n( x_1 ,. .., x_n )$ ( n = 1,2, ...) such that for any $\vartheta$ we have
$$P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} \qquad( n > n_0 ( \varepsilon, F ) )$$
Prove that
a) if F is not absolutely continuous, then the shift parameter is "well estimated",
b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".
Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.
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.)