Olympiad problems

1994 Iran MO (2nd round), 3

Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

2013 NIMO Problems, 2

Tags: quadratic , polynomial , functional equation , function , algebra

Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$. [i]Proposed by Ahaan S. Rungta[/i]

2019 Dutch IMO TST, 4

Tags: functional equation , find all functions , prime number , function , number theory

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 $p$.

2020 Durer Math Competition Finals, 5

Tags: functional equation , functional , algebra

Let $H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}$. Describe all functions $f : H \to H$ for which a) $x = f(x) - f(f(x))$ holds for every $x \in H$. b) $x = f(x) + f(f(x)) - f(f(f(x)))$ holds for every $x \in H$. c) $x = f(x) + 2f(f(x)) - 3f(f(f(x)))$ holds for every $x \in H$. PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2

1998 Brazil National Olympiad, 2

Tags: functional equation , algebra , function

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

2016 Balkan MO Shortlist, A8

Tags: function , functional equation in z , linear function , algebra , functional equation

Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.

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 $x,y\in\mathbb{Z}$.

2011 Dutch IMO TST, 2

Tags: functional equation , algebra

Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.

2015 Cuba MO, 1

Tags: functional equation , algebra , functional

Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$ (a) Find $f(1)$. (b) Find $f(x)$.

2021 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , function , substitution , functional equation

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 $y$.

2003 Bulgaria Team Selection Test, 2

Tags: functional equation , algebra

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

Mexican Quarantine Mathematical Olympiad, #5

Tags: factorial , functional equation , number theory

Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$: $$p \mid f(n)f(p-1)!+n^{f(p)}.$$ [i]Proposed by Dorlir Ahmeti[/i]

2017 Taiwan TST Round 1, 3

Tags: functional equation , algebra

Find all injective functions $ f:\mathbb{N} \to \mathbb{N} $ such that $$ f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2 $$ holds for all $ a,b \in \mathbb{N} $. Note that $ f^{k}\left(n\right) $ means $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $

2022 Indonesia MO, 1

Tags: functional equation , functional equation in r

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have \[ f(f(f(x)) + f(y)) = f(y) - f(x) \]

1997 Bosnia and Herzegovina Team Selection Test, 3

Tags: algebra , function , functional equation

It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}$$ where $c=f(1)-1$ $a)$ Solve given functional equation for $A=\mathbb{N}$ $b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$

1977 Germany Team Selection Test, 2

Tags: polynomial , algebra , functional equation

Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

PEN K Problems, 22

Tags: functional equation , induction , function

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

2019 Taiwan APMO Preliminary Test, P6

Tags: functional equation , algebra

Let $\mathbb{N}$ denote the set of all positive integers.Function $f:\mathbb{N}\cup{0}\rightarrow\mathbb{N}\cup{0}$ satisfies :for any two distinct positive integer $a,b$, we have $$f(a)+f(b)-f(a+b)=2019$$ (1)Find $f(0)$ (2)Let $a_1,a_2,...,a_{100}$ be 100 positive integers (they are pairwise distinct), find $f(a_1)+f(a_2)+...+f(a_{100})-f(a_1+a_2+...+a_{100})$

MathLinks Contest 7th, 6.2

Tags: functional equation , function , algebra

Find all functions $ f,g: \mathbb Q \to \mathbb Q$ such that for all rational numbers $ x,y$ we have \[ f(f(x) \plus{} g(y) ) \equal{} g(f(x)) \plus{} y . \]

2023 Switzerland - Final Round, 5

Tags: algebra , domain , functional equation , function

Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.

2020 Bulgaria Team Selection Test, 5

Tags: additive function , function , induction , functional equation , algebra , inequalities

Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$. Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$

2021 Science ON grade X, 4

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

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

2023 Indonesia MO, 2

Tags: functional equation , floor , algebra , floor function

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$: \[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \] [b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.

1992 IMO, 2

Tags: algebra , function , functional equation

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

2007 Stars of Mathematics, 1

Tags: functional equation , function , algebra

Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies: $ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $ $ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $ $ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $