Found problems: 1513
1993 Czech And Slovak Olympiad IIIA, 5
Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$
2002 India IMO Training Camp, 10
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
2010 Germany Team Selection Test, 3
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2012 ELMO Shortlist, 8
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
1988 China Team Selection Test, 2
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
2018 Turkey Team Selection Test, 2
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 $x,y$.
2016 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
2005 IMO Shortlist, 2
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
2024 Dutch IMO TST, 2
Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with
\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]
for all $x,y,z \in \mathbb{R}_{\ge 0}$.
2013 IMO Shortlist, N1
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 $n$.
2022 German National Olympiad, 6
Consider functions $f$ satisfying the following four conditions:
(1) $f$ is real-valued and defined for all real numbers.
(2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$.
(3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$.
(4) We have $f(2)=4$.
Prove that:
a) There is a function $f$ with $f(3)=9$ satisfying the four conditions.
b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.
2010 Victor Vâlcovici, 1
Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(2x+f(y))=x+y +f(f(x)) , \ \ \ \forall x,y \in \mathbb{R}^+.\]
1996 Estonia Team Selection Test, 3
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
1983 IMO Shortlist, 11
Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy:
\[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\]
where $b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$
2015 Abels Math Contest (Norwegian MO) Final, 1b
Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.
2001 Spain Mathematical Olympiad, Problem 6
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 $f(n) = 2001.$
2001 Dutch Mathematical Olympiad, 2
The function f has the following properties :
$f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$
$f(4) = 10$
Calculate $f(2001)$.
2016 Iran MO (3rd Round), 3
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$
$$f(y)f(x+f(y))=f(x)f(xy)$$
2017 Greece Team Selection Test, 3
Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$
and $g(1)=-8$
2007 Germany Team Selection Test, 2
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]
2025 Euler Olympiad, Round 2, 4
Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$,
$$
f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y),
$$
where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$.
[I]Proposed by Stijn Cambie, Belgium[/i]
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 $x,y \not = 0$.
1987 IMO Shortlist, 1
Let f be a function that satisfies the following conditions:
$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$.
$(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions;
$(iii)$ $f(0) = 1$.
$(iv)$ $f(1987) \leq 1988$.
$(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$.
Find $f(1987)$.
[i]Proposed by Australia.[/i]
2017 Vietnamese Southern Summer School contest, Problem 2
Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy:
$$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$
for all real numbers $x,y$.
2020 Moldova Team Selection Test, 11
Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy
$$f(\sin{x})+f(\cos{x})=2020$$
for any real number $x.$