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

2008 Korea Junior Math Olympiad, 7
Tags: function , algebra , functional equation
Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

1996 Estonia Team Selection Test, 3
Tags: function , algebra , functional equation
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 $x\ne 0$

2010 Bundeswettbewerb Mathematik, 4
Tags: polynomial , function , algebra , functional equation , functional
In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

2021 Romanian Master of Mathematics Shortlist, A1
Tags: algebra , functional equation , substitution , function
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \] for all real numbers $x$ and $y$.

2021 Dutch IMO TST, 3
Tags: functional , functional equation , algebra
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

1979 IMO Longlists, 10
Tags: algebra , polynomial , vieta , functional equation
Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

2018-IMOC, N2
Tags: number theory , least common multiple , greatest common divisor , fe , functional equation
Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$ for all $x,y\in\mathbb N$.

2021-IMOC, N8
Tags: algebra , functional equation
Find all integer-valued polynomials $$f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x)$$ holds for all positive integer $x$, where $\tau (x)$ is the number of positive factors of $x$ [i]Proposed By - ckliao914[/i]

2002 USAMO, 4
Tags: functional equation
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.

1977 IMO, 3
Tags: function , algebra , functional inequality , functional equation
Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$

2024 Dutch IMO TST, 2
Tags: functional equation , algebra
Find all functions $f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that for all positive integers $m,n$ and $a$ we have a) $f(f(m)f(n))=mn$ and b) $f(2024a+1)=2024a+1$.

2022 China Team Selection Test, 3
Tags: functional equation , algebra
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$. [i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.

2007 Germany Team Selection Test, 2
Tags: function , functional equation , algebra
Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]

2018 India National Olympiad, 6
Tags: power of prime , number theory , functional equation
Let $\mathbb N$ denote set of all natural numbers and let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $\text{(a)} f(mn)=f(m).f(n)$ for all $m,n \in\mathbb{N}$; $\text{(b)} m+n$ divides $f(m)+f(n)$ for all $m,n\in \mathbb N$. Prove that, there exists an odd natural number $k$ such that $f(n)= n^k$ for all $n$ in $\mathbb{N}$.

2017 Iran Team Selection Test, 3
Tags: algebra , functional equation , function
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

2020 Peru Cono Sur TST., P2
Tags: functional equation , number theory
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

2024 Brazil Team Selection Test, 4
Tags: algebra , functional equation , function
Find all pairs of positive integers \( (a, b) \) such that \( f(x) = x \) is the only function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, \( f^n(x) \) represents the function obtained by applying \( f \) \( n \) times to \( x \). That is, \( f^1(x) = f(x) \) and \( f^{n+1}(x) = f(f^n(x))\) for all \(n \geq 1\).

2004 India IMO Training Camp, 3
Tags: trigonometry , function , functional equation , system of equations , algebra
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2007 Italy TST, 3
Tags: function , polynomial , functional equation , algebra
Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2020 European Mathematical Cup, 4
Tags: algebra , functional equation , function
Let $\mathbb{R^+}$ denote the set of all positive real numbers. Find all functions $f: \mathbb{R^+}\rightarrow \mathbb{R^+}$ such that $$xf(x + y) + f(xf(y) + 1) = f(xf(x))$$ for all $x, y \in\mathbb{R^+}.$ [i]Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj[/i]

PEN K Problems, 5
Tags: functional equation , function , arithmetic sequence
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

2011 Swedish Mathematical Competition, 6
Tags: algebra , functional , functional equation
How many functions $f:\mathbb N \to \mathbb N$ are there such that $f(0)=2011$, $f(1) = 111$, and $$f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}$$ for all non-negative integers $x$, $y$?

2021 Thailand Mathematical Olympiad, 7
Tags: algebra , functional equation
Determine all functions $f : \mathbb R^+ \to \mathbb R$ that satisfy the equation $$f(xy) = f(x)f(y)f(x+y)$$ for all positive real numbers $x$ and $y$.

2020 Indonesia MO, 3
Tags: algebra , functional equation , divisibility
The wording is just ever so slightly different, however the problem is identical. Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.

2020 Estonia Team Selection Test, 3
Tags: algebra , functional , functional equation
Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$