Found problems: 1513
2005 Thailand Mathematical Olympiad, 3
Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.
2021 Peru IMO TST, P3
Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions:
(i) $f(f(x))=x^2$ for any $x\geq 1$;
(ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$.
1. Prove that $x<f(x)<x^2$ for any $x\geq 1$.
2. Prove that there exists a function $f$ satisfies the above two conditions and the following one:
(iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.
2025 Balkan MO, 3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
[i]Proposed by Giannis Galamatis, Greece[/i]
1998 Brazil National Olympiad, 2
Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]
2001 Miklós Schweitzer, 5
Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation
$$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$
for all positive $x,y$, then
$$2f(\sqrt{xy})=f(x)+f(y)$$
for every pair $x,y$ of positive numbers.
2016 Belarus Team Selection Test, 2
Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions:
1) $f(f(x)) =xf(x)-ax$ for all real $x$
2) $f$ is not constant
3) $f$ takes the value $a$
2022 Thailand Online MO, 10
Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions.
[list=disc]
[*] $f(a)$ is not an integer for some rational number $a$.
[*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers.
[/list]
1992 IMO Longlists, 35
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.
2020 IberoAmerican, 5
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$
2018 Brazil Undergrad MO, 2
Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?
2016 Postal Coaching, 2
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
2019 Hong Kong TST, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2009 Dutch IMO TST, 4
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.
2017 Abels Math Contest (Norwegian MO) Final, 1a
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.
2016 Azerbaijan IMO TST First Round, 4
Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$
P.S: $P(y)=y^{2015}$ is also a function with power $2015$
2023 District Olympiad, P4
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]
2004 IMO Shortlist, 3
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2025 Olympic Revenge, 3
Find all $f\colon\mathbf{R}\rightarrow\mathbf{R}$ such that
\[f(f(x)f(y)) = f(x + y) + f(xy)\]
for all $x,y\in\mathbf{R}$.
2020 Swedish Mathematical Competition, 3
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
2010 Brazil National Olympiad, 1
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
2018 SIMO, Q1
Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
2002 Singapore Senior Math Olympiad, 1
Let $f: N \to N$ be a function satisfying the following:
$\bullet$ $f(ab) = f(a)f(b)$, whenever the greatest common divisor of $a$ and $b$ is $1$.
$\bullet$ $f(p + q) = f(p)+ f(q)$ whenever $p$ and $q$ are primes.
Determine all possible values of $f(2002)$. Justify your answers.
2004 Korea Junior Math Olympiad, 5
Show that there exists no function $f:\mathbb {R}\rightarrow \mathbb {R}$ that satisfies $f(f(x))-x^2+x+3=0$ for arbitrary real variable $x$.
(Same as KMO 2004 P1)
2014 Peru IMO TST, 14
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 $n$.