Olympiad problems

2015 Miklos Schweitzer, 8

Prove that all continuous solutions of the functional equation $\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$ are the constant functions.

2019 USAJMO, 2

Tags: function , algebra , functional equation

Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \] for all integers $x$. [i]Proposed by Ankan Bhattacharya[/i]

KoMaL A Problems 2019/2020, A.756

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ [i]Based on a problem of Romanian Masters of Mathematics[/i]

2018 Abels Math Contest (Norwegian MO) Final, 3b

Tags: function , functional equation , functional , algebra

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

2009 Ukraine Team Selection Test, 4

Tags: functional equation , functional , algebra

Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$

2024 PErA, P5

Tags: functional equation , function , algebra

Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(xf(x)+y^2) = x^2+yf(y) \] for any positive reals $x,y$.

2020 Thailand Mathematical Olympiad, 3

Tags: algebra , functional equation

Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation $$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$ for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$.

2017-IMOC, A7

Tags: algebra , iteration , function , functional equation

Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies \[ f^n(n) = n + k \] for all $n \in \mathbb{N}$

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 $x, y \in R$.

2004 IMO Shortlist, 3

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

2008 National Olympiad First Round, 19

Tags: function , quadratic , algebra , functional equation

Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{There is more than one solution} $

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{R}\mapsto \mathbb{R}$ satisfy: $$f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)$$ for all real numbers $x,y$.

1994 Iran MO (2nd round), 3

Tags: function , induction , functional equation

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

2011 Brazil Team Selection Test, 4

Tags: function , algebra , functional equation

Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

1989 IMO Shortlist, 10

Tags: function , algebra , functional equation , complex number

Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

1987 Traian Lălescu, 1.2

Tags: function , algebra , functional equation

Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying $$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$ [b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $ [b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $

2015 Miklos Schweitzer, 8

2019 USAJMO, 2

KoMaL A Problems 2019/2020, A.756

2018 Abels Math Contest (Norwegian MO) Final, 3b

2009 Ukraine Team Selection Test, 4

2024 PErA, P5

2020 Thailand Mathematical Olympiad, 3

2017-IMOC, A7

2010 Indonesia TST, 1

2004 IMO Shortlist, 3

2008 National Olympiad First Round, 19

2017 Vietnamese Southern Summer School contest, Problem 2

1994 Iran MO (2nd round), 3

2011 Brazil Team Selection Test, 4

1989 IMO Shortlist, 10

1987 Traian Lălescu, 1.2

2025 Taiwan TST Round 2, A

2000 Brazil Team Selection Test, Problem 2

2006 Thailand Mathematical Olympiad, 5

2021 China National Olympiad, 6

2011 Belarus Team Selection Test, 1

1994 Bulgaria National Olympiad, 2

2003 Alexandru Myller, 4

2010 Contests, 2

2014 IFYM, Sozopol, 7