Found problems: 1513
2023 Irish Math Olympiad, P4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that
$$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$
for all real numbers $x, y$ with $xy \neq 1$.
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 $n$.
2021 Baltic Way, 1
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation
$$
(f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)
$$
for all $x ,y \in \mathbb{R}$.
2022 Balkan MO Shortlist, A6
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
2014 IFYM, Sozopol, 7
Find all $f: \mathbb{N}\rightarrow \mathbb{N}$, for which
$f(f(n)+m)=n+f(m+2014)$
for $\forall$ $m,n\in \mathbb{N}$.
2019 Romanian Masters In Mathematics, 5
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]
for all real numbers $x$ and $y$.
Dumbest FE I ever created, 2.
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]
2018 ELMO Shortlist, 1
Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?
[i]Proposed by Daniel Liu[/i]
2001 IMC, 5
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that
\[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]
2015 Miklos Schweitzer, 8
Prove that all continuous solutions of the functional equation
$\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$
are the constant functions.
2008 IMO Shortlist, 1
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2}
\]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$
[i]Author: Hojoo Lee, South Korea[/i]
Russian TST 2021, P3
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
[i]Ukraine[/i]
2019 Romania Team Selection Test, 3
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
2020 Bulgaria Team Selection Test, 5
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
2023 China Team Selection Test, P22
Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$,
$$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$
2018 Abels Math Contest (Norwegian MO) Final, 4
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.
1995 IMO Shortlist, 1
Does there exist a sequence $ F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions?
[b](a)[/b] Each of the integers $ 0, 1, 2, \ldots$ occurs in the sequence.
[b](b)[/b] Each positive integer occurs in the sequence infinitely often.
[b](c)[/b] For any $ n \geq 2,$
\[ F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)).
\]
2017 Peru IMO TST, 13
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]
2013 ELMO Shortlist, 3
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
1967 IMO Longlists, 50
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2012 Swedish Mathematical Competition, 1
The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$.
2019 OMMock - Mexico National Olympiad Mock Exam, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$
[i]Proposed by Victor DomÃnguez and Pablo Valeriano[/i]
2005 IMO Shortlist, 4
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
1990 IMO, 1
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
2019 Jozsef Wildt International Math Competition, W. 27
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$