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

2006 Spain Mathematical Olympiad, 1
Tags: functional equation , real number , algebra
Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$

2010 Germany Team Selection Test, 3
Tags: function , algebra , functional equation
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

2012 Belarus Team Selection Test, 3
Tags: function , functional equation , algebra
Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$. (E. Barabanov)

2005 Portugal MO, 6
Tags: algebra , functional , functional equation , number theory
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.

2025 Macedonian Balkan MO TST, 3
Tags: functional equation , algebra
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy \[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\] for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.

2023 ELMO Shortlist, A1
Tags: polynomial , algebra , functional equation
Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

2019 Costa Rica - Final Round, 4
Tags: algebra , function , functional equation
Let $g: R \to R$ be a linear function such that $g (1) = 0$. If $f: R \to R$ is a quadratic function such what $g (x^2) = f (x)$ and $f (x + 1) - f (x - 1) = x$ for all $x \in R$. Determine the value of $f (2019)$.

2001 China Team Selection Test, 3
Tags: function , polynomial , functional equation , algebra
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

2005 Putnam, B3
Tags: function , integration , functional equation , absolute value , logarithm , algebra
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that \[ f'\left(\frac ax\right)=\frac x{f(x)} \] for all $x>0.$

2019 China Team Selection Test, 5
Tags: functional equation , function
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.

2011 IMO Shortlist, 3
Tags: function , algebra , functional equation
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

2021 Canadian Mathematical Olympiad Qualification, 1
Tags: functional equation , algebra , polynomial
Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

2008 Indonesia TST, 4
Tags: number theory , functional equation , functional , functional equation in n
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

2013 Iran Team Selection Test, 16
Tags: function , functional equation , algebra
The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. [i]Proposed by Amirhossein Gorzi[/i]

2019 Thailand TSTST, 3
Tags: functional equation , algebra , function
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$, $\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and $\text{(iii)}$ $f(1)-f(0)$ is even.

2022 SG Originals, Q3
Tags: functional equation , number theory , factorial , function
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

2015 Indonesia MO Shortlist, A4
Tags: algebra , functional equation , function
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that \[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$ and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.

2009 Dutch IMO TST, 4
Tags: functional equation , functional equation in z , algebra
Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

1992 IMO Shortlist, 6
Tags: function , algebra , functional equation
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

2014 Nordic, 1
Tags: functional equation , function , algebra
Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.

2015 Belarus Team Selection Test, 1
Tags: functional equation , algebra , functional
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

1995 China National Olympiad, 2
Tags: function , algebra , functional equation
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

2018 Thailand TSTST, 1
Tags: functional equation , algebra
Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$

2024 Macedonian Mathematical Olympiad, Problem 3
Tags: algebra , functional equation , function
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation $$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$ for any two real numbers $x$ and $y$.

2010 Saudi Arabia BMO TST, 3
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.