Found problems: 1513
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that for $\forall$ $x,y\in \mathbb{R}$ :
$f(x+f(x+y))+xy=yf(x)+f(x)+f(y)+x$.
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[ f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2. \]for all real numbers $x$ and $y$.
As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list]
[i]Evan Chen and Yang Liu[/i]
Find all polynomials $P$ with real coefficients which satisfy
\[P(x)P(x+1)=P(x^2-x+3) \quad \forall x \in \mathbb{R}\]
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that
$$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$
for all real $x, y$.
Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.
Find all possible values of $f(1995)$.
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$,
$$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$
Find all functions f : R-->R such that
f (f (x) + y^2) = x −1 + (y + 1)f (y)
holds for all real numbers x, y
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
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 $n$.
Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions:
$f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$
Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.
Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation
$$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$
[b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational.
[b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?
In the following, let $N_0$ denotes the set of non-negative integers.
Find all polynomials $P(x)$ that fulfill the following two properties:
(1) All coefficients of $P(x)$ are from $N_0$.
(2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.
A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy
\[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.
How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and
$$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$
for all non-negative integers $x$, $y$?
Let $\mathbb{R}_+$ be the set of positive real numbers. Find all functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ such that
\[f(xy + x + y) + f \left( \frac1x \right) f\left( \frac1y \right) = 1\]
for every $x$, $y\in \mathbb{R}_+$.
[i]Proposed by Li4 and Untro368.[/i]