Olympiad problems

2009 Belarus Team Selection Test, 3

a) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+1) - f(n)$ for all $n \in N$? b) Does there exist a function $f: N \to N$ such that $f(f(n))=f(n+2) - f(n)$ for all $n \in N$? I. Voronovich

2021 Austrian MO National Competition, 4

Tags: algebra , functional equation , functional

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)

PEN K Problems, 15

Tags: functional equation , function

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]

2023 SG Originals, Q4

Tags: algebra , functional equation

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

2022 Iran MO (2nd round), 2

Tags: functional equation , function , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have: $$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$

2017 IMO Shortlist, A6

Tags: functional equation , algebra

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \] [i]Proposed by Dorlir Ahmeti, Albania[/i]

2021 Balkan MO Shortlist, A1

Tags: algebra , functional equation

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that $$f(x^2 + y^2) = g(xy)$$ holds for all $x, y \in \mathbb{R}^{+}$.

1996 IMO, 3

Tags: function , number theory , algebra , functional equation

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

2013 Korea Junior Math Olympiad, 7

Tags: functional equation

Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied. i. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$. (Here, $\mathbb{N}$ is the set of all positive integers.)

2024 Philippine Math Olympiad, P1

Tags: algebra , functional equation

Let $f:\mathbb{Z}^2\rightarrow\mathbb{Z}$ be a function satisfying \[f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)\] for all integers $x$ and $y$. Can it happen that $|f(x,y)|\leq 2024$ for all $x,y\in\mathbb{Z}$?

2003 Croatia National Olympiad, Problem 2

Tags: algebra , functional equation

A sequence $(a_n)_{n\ge0}$ satisfies $a_{m+n}+a_{m-n}=\frac12\left(a_{2m}+a_{2n}\right)$ for all integers $m,n$ with $m\ge n\ge0$. Given that $a_1=1$, find $a_{2003}$.

2014 International Zhautykov Olympiad, 2

Tags: function , functional equation , algebra

Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

2024 Argentina Iberoamerican TST, 5

Tags: functional equation , algebra

Let $ \mathbb R $ be the set of real numbers. Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that, for all real numbers $ x $ and $ y $, the following equation holds:$$\big (x^2-y^2\big )f\big (xy\big )=xf\big (x^2y\big )-yf\big (xy^2\big ).$$

2008 Dutch IMO TST, 1

Tags: functional equation , functional equation in n , algebra

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

1993 IMO Shortlist, 6

Tags: function , algebra , functional equation

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

2021 Romania National Olympiad, 4

Tags: functional equation , algebra

Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that \[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\] Also, determine all pairs of functions with this property. [i]Vasile Pop[/i]

2022 Francophone Mathematical Olympiad, 1

Tags: functional equation , function , algebra

find all functions $f:\mathbb{Z} \to \mathbb{Z} $ such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$

2020 Final Mathematical Cup, 1

Tags: algebra , functional equation in r , functional equation

Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true. $$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$

2019 Iran Team Selection Test, 5

Tags: function , algebra , 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]

2025 Kosovo National Mathematical Olympiad`, P4

Tags: number theory , gcd , functional equation , solve in natural , prime

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.

2021 Korea - Final Round, P6

Tags: function , functional equation

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

2023 Brazil EGMO Team Selection Test, 1

Tags: increasing function , strictly increasing , functional equation , number theory , algebra , function

Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.

1957 Miklós Schweitzer, 5

Tags: functional equation

[b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases: (a) $x,y,z$ are arbitrary non-zero real numbers; (b) $a<x,y,z<b (1<a^{3}<b)$. [b](R. 13)[/b]

2010 Indonesia TST, 1

Tags: algebra , functional , functional equation

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

2020-IMOC, A1

Tags: inequalities , am-gm , algebra , functional equation

$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$. [i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b]. [color=#B6D7A8]#1733[/color]