Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

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 \[ f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr). \]

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]

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]

find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st $f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$ related: https://artofproblemsolving.com/community/c6h381298

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

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 $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that: \[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \] where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[xf(x+f(y))=(y-x)f(f(x)).\] [i]Proposed by Nikola Velov, Macedonia[/i]

$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation $f(xy+x+y)=f(xy)+f(x)+f(y)$ is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.

Find all functions $f :[0, +\infty) \rightarrow [0, +\infty)$ for which $f(f(x)+f(y)) = xy f (x+y)$ for every two non-negative real numbers $x$ and $y$.

Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

Function $f:\mathbb{R}\to\mathbb{R}$, satisfies that $f(0)=1$, and $f(xy+1)=f(x)f(y)-f(y)-x+2$, then $f(x)=$________.

Find all functions $f : R \to R$ such that for all $x, y \in R$: $$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

Let $N$ be the set of positive integers. Find all functions $f : N \to N$ such that both $\bullet$ $f(f(m)f(n)) = mn$ $\bullet$ $f(2022a + 1) = 2022a + 1$ hold for all positive integers $m, n$ and $a$.

Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and \[ f(x) - f(y) \leq (x-y)(g(x) - g(y)) \] for all real numbers $x$ and $y$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.