Olympiad problems

2011 IMAR Test, 3

Given an integer number $n \ge 2$, show that there exists a function $f : R \to R$ such that $f(x) + f(2x) + ...+ f(nx) = 0$, for all $x \in R$, and $f(x) = 0$ if and only if $x = 0$.

2020 Miklós Schweitzer, 8

Tags: algebra , functional equation , number theory , function

Let $\mathbb{F}_{p}$ denote the $p$-element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$. Find all functions $D\colon S\to S$ satisfying \[D(f\circ g)=(D(f)\circ g)\cdot D(g)\] for all $f,g \in {S}$. Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$, and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$.

2016 Greece Team Selection Test, 3

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

2022 Korea -Final Round, P3

Tags: functional equation , algebra

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

2005 Miklós Schweitzer, 8

Tags: real analysis , functional equation , algebra

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ [hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]

2013 Saudi Arabia Pre-TST, 3.1

Tags: functional , functional equation , algebra

Let $f : R \to R$ be a function satisfying $f(f(x)) = 4x + 1$ for all real number $x$. Prove that the equation $f(x) = x$ has a unique solution.

2021 Dutch IMO TST, 3

Tags: functional , algebra , functional equation

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

2023 District Olympiad, P4

Tags: number theory , functional equation

Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.

2013 Costa Rica - Final Round, F1

Tags: functional , algebra , functional equation

Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$

PEN K Problems, 11

Tags: algebra , functional equation , function

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]

2014 Contests, 2

Tags: algebra , function , limit , functional equation

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

1985 Vietnam Team Selection Test, 3

Tags: algebra , function , functional equation

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

1987 IMO Shortlist, 22

Tags: algebra , functional equation , function

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

2025 Balkan MO, 3

Tags: algebra , functional equation

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

2020 Azerbaijan IZHO TST, 3

Tags: functional equation , algebra

Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$ for all $x,y\in{\mathbb{R}}$

2023 Serbia National Math Olympiad, 5

Tags: algebra , calculus , function , functional equation

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.

2024 Chile TST IMO, 4

Tags: functional equation , algebra

Let $\alpha$ be a real number. Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x+y))=f(x+y) +f(x)f(y)+ \alpha xy$ for all $x,y \in \mathbb{R}$

2021 Balkan MO Shortlist, A5

Tags: functional equation , algebra , shortlist

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(xf(x + y)) = yf(x) + 1$$ holds for all $x, y \in \mathbb{R}^{+}$. [i]Proposed by Nikola Velov, North Macedonia[/i]

2021 Science ON grade X, 2

Tags: functional equation , combinatorics , algebra

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

2005 Taiwan TST Round 2, 1

Tags: function , algebra , functional equation

Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$. This is much harder than the problems we had in the 1st TST...

2007 IMO Shortlist, 4

Tags: functional equation , algebra

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/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 $x, y \in \mathbb{R}^{+}$.

2010 Contests, 1

Tags: algebra , functional equation , fe

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

1990 IMO Longlists, 93

Tags: algebra , function , functional equation , number theory

Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that \[ f(xf(y)) \equal{} \frac {f(x)}{y} \] for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.

2018 Peru EGMO TST, 2

Tags: algebra , functional equation

Find all functions $f:\mathbb R \rightarrow \mathbb R$, such that $2xyf(x^2-y^2)=(x^2-y^2)f(x)f(2y)$