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

2003 China Team Selection Test, 2
Tags: function , algebra , functional equation
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

1984 Austrian-Polish Competition, 9
Tags: algebra , functional equation , functional
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$

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

2015 Saudi Arabia GMO TST, 1
Tags: functional , algebra , functional equation
Find all functions $f : R \to R$ satisfying the following conditions (a) $f(1) = 1$, (b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$ (c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$ Trần Nam Dũng

2015 Thailand TSTST, 1
Tags: functional equation , algebra
Let $A$ and $B$ be nonempty sets and let $f : A \to B$. Prove that the following statements are equivalent: $\text{(i) }$ $f$ is surjective. $\text{(ii)} $ For every set $C$ and and every functions $g, h : B \to C$, if $g\circ f = h \circ f$ then $g = h$.

1989 Romania Team Selection Test, 1
Tags: algebra , functional , functional equation
Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

2011 International Zhautykov Olympiad, 2
Tags: algebra , function , functional equation
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality, \[f(x+f(y))=f(x-f(y))+4xf(y)\] for any $x,y\in\mathbb{R}$.

2011 Postal Coaching, 3
Tags: function , algebra , functional equation
Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that \[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\] for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.

2025 Korea - Final Round, P2
Tags: functional equation , function , algebra
Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$ [b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$

2018 Iran MO (3rd Round), 2
Tags: functional equation , algebra
Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that: $f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$

2024 Romania EGMO TST, P1
Tags: functional equation , function , algebra
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

2008 Indonesia TST, 4
Tags: functional equation , functional , functional equation in n , number theory
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 $f(p)|f(n)$.

2016 India IMO Training Camp, 2
Tags: algebra , functional equation , function
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.)

MathLinks Contest 5th, 2.1
Tags: functional equation
For what positive integers $k$ there exists a function $f : N \to N$ such that for all $n \in N$ we have $\underbrace{\hbox{f(f(... f(n)....))}}_{\hbox{k times}} = f(n) + 2$ ?

2018 Estonia Team Selection Test, 4
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

2016 Iran MO (3rd Round), 2
Tags: function , algebra , functional equation , number theory
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ , $(f(a)+b) f(a+f(b))=(a+f(b))^2$

2017 Dutch IMO TST, 4
Tags: functional equation , algebra
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

1998 AMC 12/AHSME, 17
Tags: algebra , function , functional equation , 3d geometry , geometry
Let $ f(x)$ be a function with the two properties: [list=a] [*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and [*] $ f(0) \equal{} 2$ [/list] What is the value of $ f(1998)$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 1996\qquad \textbf{(D)}\ 1998\qquad \textbf{(E)}\ 2000$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3
Tags: algebra , functional equation , functional
Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.

2007 Singapore MO Open, 4
Tags: algebra , functional equation
find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st $f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$ related: https://artofproblemsolving.com/community/c6h381298

1988 IMO Shortlist, 19
Tags: algebra , functional equation , function
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

1996 Singapore Team Selection Test, 2
Tags: functional equation , functional , algebra
Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.

2021 China National Olympiad, 6
Tags: functional equation
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$

2025 Bangladesh Mathematical Olympiad, P10
Tags: algebra , functional equation
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$ for all $x, y \in \mathbb{R}$. [i]Proposed by Md. Fuad Al Alam[/i]

2008 Indonesia TST, 2
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.