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

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Found problems: 1513

2016 Romania Team Selection Test, 2
Tags: function , number theory , functional equation , algebra
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

2013 Macedonian Team Selection Test, Problem 3
Tags: function , number theory , functional equation , algebra
Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that: $(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ [b]min[/b] $\left \{ f(a),f(b) \right \}$. $(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.

2024 Kyiv City MO Round 1, Problem 5
Tags: functional equation , number theory
Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

2020 GQMO, 4
Tags: functional equation , algebra
For all real numbers $x$, we denote by $\lfloor x \rfloor$ the largest integer that does not exceed $x$. Find all functions $f$ that are defined on the set of all real numbers, take real values, and satisfy the equality \[f(x + y) = (-1)^{\lfloor y \rfloor} f(x) + (-1)^{\lfloor x \rfloor} f(y)\] for all real numbers $x$ and $y$. [i]Navneel Singhal, India[/i]

2004 Thailand Mathematical Olympiad, 2
Tags: functional equation , algebra , functional
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.

2011 Indonesia TST, 1
Tags: functional , algebra , functional equation , functional equation in q
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

1985 Traian Lălescu, 1.3
Tags: function , algebra , functional equation , polynomial
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that $$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$ for all integer polynomials $ p. $

2017 Abels Math Contest (Norwegian MO) Final, 1b
Tags: algebra , function , functional equation
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

2013 Brazil Team Selection Test, 4
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.

2009 Belarus Team Selection Test, 3
Tags: functional equation , algebra , functional
Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$. I.Voronovich

1995 IMO Shortlist, 1
Tags: functional equation , iteration , algebra
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,$ \[ F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)). \]

2024 Baltic Way, 1
Tags: algebra , functional equation
Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that \[ xf(x+y)=(x+\alpha y)f(x)+xf(y) \] for all $x,y\in\mathbb{R}$.

2013 IMO Shortlist, A3
Tags: function , functional equation , algebra
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

2018-IMOC, N1
Tags: number theory , fe , functional equation
Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.

2025 Korea Winter Program Practice Test, P1
Tags: functional equation , algebra
Determine all functions $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for any positive reals $x,y$, $$f(xy+f(xy)) = xf(y) + yf(x)$$

2013 Romanian Master of Mathematics, 2
Tags: floor function , function , algebra , induction , functional equation
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?

2000 Switzerland Team Selection Test, 12
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$

2023 Mexican Girls' Contest, 4
Tags: functional equation , algebra
A function $g$ is such that for all integer $n$: $$g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}$$ A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$: $$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$ $$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$ Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$

2019 Hong Kong TST, 1
Tags: functional equation , algebra
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 $x,y\in\mathbb{Q}_{>0}$

1991 Irish Math Olympiad, 4
Tags: functional equation , functional equation in q , algebra
Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$. Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.

1967 Putnam, A4
Tags: function , functional equation , integral
Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

2005 Slovenia Team Selection Test, 2
Tags: algebra , functional , functional equation
Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

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 $x, y \in Z$

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

2019 Iran Team Selection Test, 5
Tags: function , functional equation , algebra
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]