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

2019 IMO Shortlist, A1
Tags: functional equation
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

2023 Israel TST, P1
Tags: algebra , function , functional equation
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds: \[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]

2011 Belarus Team Selection Test, 1
Tags: functional equation , trigonometry , functional , algebra
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

2016 Azerbaijan Team Selection Test, 3
Tags: functional equation , algebra , function
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.

2019 Iran Team Selection Test, 5
Tags: algebra , function , functional equation
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]

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 $y$.

2018 Switzerland - Final Round, 5
Tags: algebra , function , functional equation
Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true : $$f(xf(x)+yf(y)) = xy$$

2021 Final Mathematical Cup, 1
Tags: functional equation , functional , algebra
Let $N$ is the set of all positive integers. Determine all mappings $f: N-\{1\} \to N$ such that for every $n \ne m$ the following equation is true $$f(n)f(m)=f\left((nm)^{2021}\right)$$

2014 Contests, 4
Tags: functional equation , algebra
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$

2019 239 Open Mathematical Olympiad, 6
Tags: algebra , functional equation
Find all functions $f : (0, +\infty) \to \mathbb{R}$ satisfying the following conditions: $(i)$ $f(x) + f(\frac{1}{x}) = 1$ for all $x> 0$; $(ii)$ $f(xy + x + y) = f(x)f(y)$ for all $x, y> 0$.

1994 French Mathematical Olympiad, Problem 5
Tags: functional equation , algebra , fe
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

2004 Korea Junior Math Olympiad, 5
Tags: functional equation , algebra , existence , function
Show that there exists no function $f:\mathbb {R}\rightarrow \mathbb {R}$ that satisfies $f(f(x))-x^2+x+3=0$ for arbitrary real variable $x$. (Same as KMO 2004 P1)

1995 IMO Shortlist, 5
Tags: function , algebra , functional equation
Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

2019 Dutch IMO TST, 4
Tags: number theory , function , find all functions , prime number , functional equation
Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2014 Middle European Mathematical Olympiad, 1
Tags: function , functional equation , fe , real , algebra
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

2011 VJIMC, Problem 4
Tags: polynomial , fe , irreducibility , functional equation , algebra
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.

2024 India IMOTC, 19
Tags: functional equation , algebra
Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\ [color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\] [color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]} f(X) = 2^{n+1}-1.\] (Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\ [i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]

2003 Alexandru Myller, 4
Tags: functional equation , real analysis , function , find all functions
Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

2023 APMO, 4
Tags: functional equation , fe
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

2019 Philippine TST, 2
Tags: algebra , functional equation
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation $$f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}$$ for all real numbers $x$ and $y$.

2011 Czech and Slovak Olympiad III A, 6
Tags: functional equation , function , algebra
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$, we have \[ f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.\]

2018 Baltic Way, 4
Tags: algebra , functional equation
Find all functions $f:[0, \infty) \to [0,\infty)$, such that for any positive integer $n$ and and for any non-negative real numbers $x_1,x_2,\dotsc,x_n$ \[f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.\]

2022 Moldova Team Selection Test, 5
Tags: algebra , function , functional equation
The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies: $1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$ $2) f(20^{22})=f(22^{20});$ $3) f(2021)=2022$. Find all possible values of $f(2022)$.

1991 Putnam, B2
Tags: algebra , superior algebra , function , functional equation
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

2017 Canadian Mathematical Olympiad Qualification, 3
Tags: algebra , functional equation , function , difference of squares , special factorizations
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$