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

1999 Austrian-Polish Competition, 3
Tags: system of equations , functional equation , functional , algebra
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

2014 BMT Spring, 8
Tags: functional equation , fe , algebra
Suppose an integer-valued function $f$ satisfies $$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$ for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.

2017 Taiwan TST Round 2, 1
Tags: algebra , functional equation
Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $

2007 Korea Junior Math Olympiad, 6
Tags: bijective function , functional equation , function , algebra
Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satis es the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

2019 Romania Team Selection Test, 3
Tags: functional equation , algebra , functional equation in n
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

1981 IMO, 3
Tags: function , functional equation , algebra
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

2025 Nordic, 1
Tags: algebra , function , fe , functional equation
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying: $(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

2016 239 Open Mathematical Olympiad, 7
Tags: algebra , functional equation
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$ for all $x>0, y>x+1$.

2021 Federal Competition For Advanced Students, P2, 4
Tags: functional equation , functional , algebra
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

2023 Iran Team Selection Test, 3
Tags: function equation , algebra , functional equation
Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$. [i]Proposed by Amirhossein Zolfaghari [/i]

2015 Ukraine Team Selection Test, 8
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$

2004 National High School Mathematics League, 8
Tags: algebra , functional equation , function
Function $f:\mathbb{R}\to\mathbb{R}$, satisfies that $f(0)=1$, and $f(xy+1)=f(x)f(y)-f(y)-x+2$, then $f(x)=$________.

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.

2018 Federal Competition For Advanced Students, P2, 1
Tags: algebra , functional equation
Let $a \ne 0$ be a real number. Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$ for all $x, y \in R_{>0}$. [i](Proposed by Walther Janous)[/i]

2014 Belarus Team Selection Test, 2
Tags: number theory , functional equation , algebra
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2008 Indonesia TST, 2
Tags: algebra , functional equation , functional
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 $x, y$.

2016 Taiwan TST Round 1, 4
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 $x,y\in\mathbb{Z}$.

2022 Iran MO (2nd round), 2
Tags: algebra , functional equation , function
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have: $$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$

PEN K Problems, 9
Tags: function , functional equation
Find all functions $f: \mathbb{N}_{0}\rightarrow \mathbb{N}_{0}$ such that for all $n\in \mathbb{N}_{0}$: \[f(f(n))+f(n)=2n+6.\]

1999 IMO Shortlist, 5
Tags: algebra , functional equation
Find all the functions $f: \mathbb{R} \to\mathbb{R}$ such that \[f(x-f(y))=f(f(y))+xf(y)+f(x)-1\] for all $x,y \in \mathbb{R} $.

2007 Grigore Moisil Intercounty, 3
Tags: function , injectivity , algebra , functional equation
Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.

2019 IMO Shortlist, A7
Tags: functional equation , arithmetic sequence , algebra
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying \[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\] for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set \[X_v=\{x\in\mathbb Z:f(x)=v\}\] is finite and nonempty. (a) Prove that there exists such a function $f$ for which there is an $f$-rare integer. (b) Prove that no such function $f$ can have more than one $f$-rare integer. [i]Netherlands[/i]

2017 Thailand TSTST, 4
Tags: function , functional equation , algebra
Find all function $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ that satisfy: $(f(1))^3+(f(2))^3+...+(f(n))^3=(f(1)+f(2)+...+f(n))^2$

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 $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

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 $y$.