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

1994 Swedish Mathematical Competition, 6
Tags: number theory , functional , functional equation
Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

2020 Dutch IMO TST, 3
Tags: functional equation , functional equation in z , functional , algebra
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

2016 India IMO Training Camp, 2
Tags: functional equation , function , algebra
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

2009 Germany Team Selection Test, 2
Tags: function , number theory , modular arithmetic , divisor , functional equation
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

2016 Germany Team Selection Test, 2
Tags: functional equation
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

2016 Latvia Baltic Way TST, 4
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

1994 Austrian-Polish Competition, 1
Tags: functional , functional inequality , functional equation , algebra
A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

Revenge EL(S)MO 2024, 4
Tags: algebra , functional equation
Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \] for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by [i]Benny Wang[/i]

2018 Swedish Mathematical Competition, 2
Tags: algebra , functional equation , functional
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$. (Here $\sqrt[3]{x}$ is defined all over $R$.)

1968 German National Olympiad, 3
Tags: algebra , functional , functional equation
Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

2012 Turkey MO (2nd round), 3
Tags: function , functional equation , algebra
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.

2019 Baltic Way, 3
Tags: algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(xf(y)-y^2)=(y+1)f(x-y)$$ holds for all $x,y\in\mathbb{R}$.

2019 Germany Team Selection Test, 1
Tags: functional equation , algebra
Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

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

2021 Bangladesh Mathematical Olympiad, Problem 12
Tags: functional equation , algebra
A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

2002 Austrian-Polish Competition, 7
Tags: function , algebra , functional equation
Find all real functions $f$ definited on positive integers and satisying: (a) $f(x+22)=f(x)$, (b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$ for all positive integers $x$ and $y$.

1993 Tournament Of Towns, (377) 5
Tags: function , functional , functional equation , algebra
Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

2019 Romanian Master of Mathematics, 5
Tags: fe , algebra , functional equation
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 $y$.

2019-IMOC, A5
Tags: functional equation , positive integer fe , iterated fe , algebra
Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity $$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$ holds for all $x,y \in \mathbb N$

2020 GQMO, 5
Tags: functional equation , algebra
Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

1998 Nordic, 1
Tags: functional equation , rational function , function , algebra
Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

2004 Austrian-Polish Competition, 7
Tags: algebra , functional equation , relatively prime , number theory
Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers: \[f(x+y) = f(x+1) + f(y+1).\]

2020 Azerbaijan IZHO TST, 3
Tags: algebra , functional equation
Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$ for all $x,y\in{\mathbb{R}}$

2000 Slovenia National Olympiad, Problem 2
Tags: functional equation , fe
Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$, $$f(x-f(y))=1-x-y.$$

1980 IMO Longlists, 7
Tags: function , algebra , functional equation
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.