Found problems: 1513
Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.
It is given the function $f:\mathbb{R}\rightarrow \mathbb{R}$ fow which $f(1)=1$ and for all $x\in\mathbb{R}$ satisfied
$f(x+5)\geq f(x)+5$ and $f(x+1)\leq f(x)+1$
If $g(x)=f(x)-x+1$ then find $g(2016)$ .
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ :
$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$.
[i]Proposed by Amirhossein Zolfaghari [/i]
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$.
[i]Proposed by Netherlands[/i]
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]
Let $S$ be a finite set of nonzero real numbers, and let $f : S\to S$ be a function with the following property:
for each $x \in S$, either $f ( f (x)) = x+ f (x)$ or $f ( f (x)) = \frac{x+ f (x)}{2}$.
Prove that $f (x) = x$ for all $x \in S$.
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that \[ f(x^2 + yf(z)) = xf(x) + zf(y) , \] for all $x, y, z \in \mathbb{R}$.
Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions:
1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$;
2. $f$ takes all integer values.
[i](I. Voronovich)[/i]
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$
holds for all $x,y\in\mathbb{R^{+}}.$
($f^m(x) =$ $f$ applies $m$ times.)
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.
$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$?
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,
$$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$.
(b) Calculate $f(n)$ for any natural number $n$.
Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that
$$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]
[i]Proposed by Aritra Mondal[/i]
Let $f^{(n)}(x)$ denote the $n^{\text{th}}$ iterate of function $f$, i.e $f^{(1)}(x)=f(x)$, $f^{(n+1)}(x)=f(f^{(n)}(x))$.
Let $p(n)$ be a given polynomial with integer coefficients, which maps the positive integers into the positive integers. Is it possible that the functional equation $f^{(n)}(n)=p(n)$ has exactly one solution $f$ that maps the positive integers into the positive integers?
[i]Submitted by Dávid Matolcsi and Kristóf Szabó, Budapest[/i]
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 $n$.
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy
$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$
for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.
[i]L. Davidov[/i]
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.
Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$