Found problems: 1513
2000 Belarus Team Selection Test, 4.1
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
2023 IMC, 3
Find all polynomials $P$ in two variables with real coefficients satisfying the identity
$$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$
2002 IMO Shortlist, 1
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
KoMaL A Problems 2021/2022, A. 825
Find all functions $f:\mathbb Z^+\to\mathbb R^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to\infty}\dfrac{f(n+1)}{f(n)}=1$.
2015 Middle European Mathematical Olympiad, 2
Determine all functions $f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}$ such that
$$f(x^2yf(x))+f(1)=x^2f(x)+f(y)$$
holds for all nonzero real numbers $x$ and $y$.
2021-IMOC, A4
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
2015 Ukraine Team Selection Test, 8
Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$
The Golden Digits 2024, P1
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties:
1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$.
2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$.
[i]Proposed by Pavel Ciurea[/i]
2012 IMO, 4
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
[i]Proposed by Liam Baker, South Africa[/i]
2017 Peru IMO TST, 1
Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
\[ f(xy-1) + f(x)f(y) = 2xy-1 \]
for all x and y
1996 IMO Shortlist, 7
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
2004 Olympic Revenge, 4
Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.
1981 IMO, 3
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
1987 IMO, 1
Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.
2016 239 Open Mathematical Olympiad, 7
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$ for all $x>0, y>x+1$.
2011 USA TSTST, 1
Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
2022 Abelkonkurransen Finale, 4a
Find all functions $f:\mathbb R^+ \to \mathbb R^+$ satisfying
\begin{align*}
f\left(\frac{1}{x}\right) \geq 1 - \frac{\sqrt{f(x)f\left(\frac{1}{x}\right)}}{x} \geq x^2 f(x),
\end{align*}
for all positive real numbers $x$.
2003 Olympic Revenge, 6
Find all functions $f:R^{*} \rightarrow R$ such that $f(x)\not = x$ and
$$ f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x} $$ for any $x,y \not = 0$.
2022 Saudi Arabia IMO TST, 3
Find all non-constant functions $f : Q^+ \to Q^+$ satisfying the equation $$f(ab + bc + ca) =f(a)f(b) +f(b)f(c)+f(c)f(a)$$
for all $a, b,c \in Q^+$ .
2015 Costa Rica - Final Round, F2
Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$.
2014 France Team Selection Test, 4
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 $n$.
PEN K Problems, 29
Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$:
\[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]
2022 Pan-African, 4
Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold
$$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$
$$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$
[i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]
1997 Romania National Olympiad, 1
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon:
1- $f(0,x)=x+1$
2- $f(x+1,0)=f(x,1)$
3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997)
find $f(3,1997)$
2013 Stars Of Mathematics, 4
Given a (fixed) positive integer $N$, solve the functional equation
\[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\]
[i](Dan Schwarz)[/i]