Olympiad problems

2013 Albania Team Selection Test, 3

Solve the function $f: \Re \to \Re$: \[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]

1993 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: functional , functional equation , number theory

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$. (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

1980 IMO Shortlist, 7

Tags: functional equation , function , algebra

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

2014 France Team Selection Test, 4

Tags: algebra , functional equation , number theory

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 Bulgarian Autumn Math Competition, Problem 12.3

Tags: functional equation in r , functional equation , algebra , function

Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]

2009 Middle European Mathematical Olympiad, 1

Tags: functional equation , real , algebra , function

Find all functions $ f: \mathbb{R} \to \mathbb{R}$, such that \[ f(xf(y)) \plus{} f(f(x) \plus{} f(y)) \equal{} yf(x) \plus{} f(x \plus{} f(y))\] holds for all $ x$, $ y \in \mathbb{R}$, where $ \mathbb{R}$ denotes the set of real numbers.

2019 IMO, 1

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]

2001 Saint Petersburg Mathematical Olympiad, 11.6

Tags: algebra , function , functional equation

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that for any $x,y$ the following is true: $$f(x+y+f(y))=f(x)+2y$$ [I]proposed by F. Petrov[/i]

2004 Germany Team Selection Test, 2

Tags: functional equation , function , algebra

Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties: (a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$. (b) We have $f\left(2\right) = 0$. (c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$. [b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.

2021 Kazakhstan National Olympiad, 5

Tags: algebra , functional equation

Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$ for all $x, y\in \mathbb{R^{+}}$

2005 India IMO Training Camp, 2

Tags: divisibility , functional equation , function , number theory , algebra

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

1984 Czech And Slovak Olympiad IIIA, 6

Tags: functional equation , algebra , function

Let f be a function from the set Z of all integers into itself, that satisfies the condition for all $m \in Z$, $$f(f(m)) =-m. \ \ (1)$$ Then: (a) $f$ is a mutually unique mapping, i.e. a simple mapping of the set $Z$ onto the set $Z$ , (b) for all $m \in Z$ holds that $f(-m) = -f(m)$ , (c) $f(m) = 0$ if and only if $m = 0$ . Prove these statements and construct an example of a mapping f that satisfies condition (1).

2008 Indonesia TST, 2

Tags: functional , algebra , functional equation

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 $x, y$.

2022 Switzerland Team Selection Test, 12

Tags: functional equation , algebra

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$

2016 Balkan MO Shortlist, A6

Tags: algebra , function , functional equation

Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that: $f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$ by Greece, Athanasios Kontogeorgis (aka socrates)

2020 Iranian Our MO, 6

Tags: algebra , functional equation , polynomial

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]

2021 Bulgaria National Olympiad, 3

Tags: algebra , functional equation

Find all $f:R^+ \rightarrow R^+$ such that $f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$ @below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url] [quote]Feel free to start individual threads for the problems as usual[/quote]

1959 Putnam, A3

Tags: function , complex , functional equation

Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$ for all $z\in \mathbb{C}$.

2017 OMMock - Mexico National Olympiad Mock Exam, 5

Tags: function , functional equation , algebra

Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation $$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$ for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$. [i]Proposed by Maximiliano Sánchez[/i]

2013 Saudi Arabia GMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$ and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$

2023 Iran Team Selection Test, 3

Tags: algebra , function equation , 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]

2024 Switzerland Team Selection Test, 12

Tags: integer , functional equation

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

1997 Romania National Olympiad, 1

Tags: algebra , function , induction , functional equation

function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

1992 IMO Shortlist, 2

Tags: recurrence relation , functional equation , quadratic , algebra

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.

1987 IMO Longlists, 6

Tags: function , functional equation , algebra

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]