Found problems: 2349
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\] for each $x,y\in\mathbb Z_{\ge 0}$.
[i]Andrei Bâra[/i]
Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(yf(x))+f(x-1)=f(x)f(y)$ and $|f(x)|<2022$ for all $0<x<1$.
A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.
Find all functions $f : R \to R$ satisfying $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that
$\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$;
$\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$.
Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and
max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$
Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.
Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify
$$ f(x+a+f(y))=f(x+b)+y, $$
for any rational $ x,y. $
[i]Vasile Pop[/i]
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]
Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$
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 $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$
[i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
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]
Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
Find all functions $f: \mathbb{Q}^{+}\to \mathbb{Q}^{+}$ such that for all $x,y \in \mathbb{Q}$: \[f \left( x+\frac{y}{x}\right) =f(x)+\frac{f(y)}{f(x)}+2y, \; x,y \in \mathbb{Q}^{+}.\]
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$
Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .
Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following:
$$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$
[i]Proposed by Vadym Koval[/i]
Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that:
$ f(x)+f(\frac1x)=1$
$ 2f(f(x))=f(2x)$
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true:
$$f(x+y+f(y))=f(x)+2y$$
[I]proposed by F. Petrov[/i]