This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1513

1996 Czech And Slovak Olympiad IIIA, 5
Tags: functional , functional equation , algebra
For which integers $k$ does there exist a function $f : N \to Z$ such that $f(1995) =1996$ and $f(xy) = f(x)+ f(y)+k f(gcd(x,y))$ for all $x,y \in N$?

2013 Costa Rica - Final Round, F1
Tags: algebra , functional equation , functional
Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

2009 Thailand Mathematical Olympiad, 5
Tags: algebra , functional equation , functional
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

2024 Thailand Mathematical Olympiad, 3
Tags: functional equation , function , algebra
Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.

2019 Saudi Arabia Pre-TST + Training Tests, 4.2
Tags: functional , functional equation , algebra
Find all functions $f : R^2 \to R$ that for all real numbers $x, y, z$ satisfies to the equation $f(f(x,z), f(z, y))= f(x, y) + z$

2015 Turkey Team Selection Test, 7
Tags: functional equation , algebra , function , sophie germain identity
Find all the functions $f:R\to R$ such that \[f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))\] for every real $x,y$.

2023 ISL, N8
Tags: integer , functional equation
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

2023 Costa Rica - Final Round, 3.1
Tags: functional equation , algebra , nonnegative integers
Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have \[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\] Determine the value of $f(2023)/f(2022)$.

2020-IMOC, N5
Tags: functional equation , number theory
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

2001 Iran MO (3rd Round), 1
Tags: function , induction , number theory , functional equation
Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that: $ f(x)+f(\frac1x)=1$ $ 2f(f(x))=f(2x)$

2022 Auckland Mathematical Olympiad, 9
Tags: number theory , functional equation , functional , algebra
Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfi ed $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$

PEN K Problems, 7
Tags: functional equation , function
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(n))+f(n)=2n+2001 \text{ or }2n+2002.\]

2018 Middle European Mathematical Olympiad, 1
Tags: algebra , function , functional equation
Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$ for all $x,y\in Q^+ .$

1994 Italy TST, 3
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.

2023 Israel TST, P3
Tags: functional equation , algebra , function
Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\] holds for any $x,y\in \mathbb{Z}$.

1999 Switzerland Team Selection Test, 3
Tags: functional equation , functional , algebra
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.

2024 IFYM, Sozopol, 5
Tags: algebra , functional equation
The function $f: A \rightarrow A$ is such that $f(x) \leq x^2 \mbox{ and } f(x+y) \leq f(x) + f(y) + 2xy$ for any $x, y \in A$. a) If $A = \mathbb{R}$, find all functions satisfying the conditions. b) If $A = \mathbb{R}^{-}$, prove that there are infinitely many functions satisfying the conditions. [i](With $\mathbb{R}^{-}$ we denote the set of negative real numbers.)[/i]

2024 Austrian MO National Competition, 1
Tags: functional equation , parameter , parameterization , algebra
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

2006 QEDMO 2nd, 13
Tags: algebra , functional , functional equation
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any two reals $x$ and $y$, we have $f\left( f\left( x+y\right) \right) +xy=f\left( x+y\right) +f\left( x\right) f\left( y\right) $.

1998 IMO Shortlist, 5
Tags: number theory , prime number , algebra , functional equation
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

2021 USAJMO, 1
Tags: function , functional equation , obi-wan kenobi , qui-gon jinn , darth sidious , daddy diddy
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]

PEN K Problems, 1
Tags: functional equation , function
Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.

2014 Serbia National Math Olympiad, 1
Tags: function , functional equation
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$ [i]Proposed by Dusan Djukic[/i]

2014 Turkey Team Selection Test, 2
Tags: function , symmetry , algebra , functional equation
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

2019 Belarusian National Olympiad, 11.7
Tags: functional equation , function , algebra
Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality $$ f(f(x)+f(y))=(x+y)f(x+y) $$ for all real $x$ and $y$. [i](B. Serankou)[/i]