Olympiad problems

1981 Romania Team Selection Tests, 4.

Determine the function $f:\mathbb{R}\to\mathbb{R}$ such that $\forall x\in\mathbb{R}$ \[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\] and draw its graph. Find all $k\in\mathbb{R}$ for which the equation $f(x)+mx+k=0$ has solutions for any $m\in\mathbb{R}$. [i]V. Preda and P. Hamburg[/i]

2010 Indonesia TST, 2

Tags: algebra , functional equation , function , induction

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$. [i]Hery Susanto, Malang[/i]

1998 Belarus Team Selection Test, 1

Tags: algebra , functional , periodic , functional equation

Do there exist functions $f : R \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$ for all real $x$ ?

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 $x, y \in R$.

2017 EGMO, 2

Tags: algebra , function , coloring , functional equation

Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties: $(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$ $(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$ [i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]

1996 Austrian-Polish Competition, 8

Tags: functional , polynomial , algebra , functional equation

Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.

1987 IMO Longlists, 74

Tags: functional equation , algebra , 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]

2015 Belarus Team Selection Test, 3

Tags: functional equation , function , algebra

Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$. [i]Proposed by Netherlands[/i]

2019 Singapore MO Open, 2

Tags: algebra , functional equation , function

find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$

2013 USA Team Selection Test, 4

Tags: combinatorics , induction , functional equation , arrows , algebra , function

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function, and let $f^m$ be $f$ applied $m$ times. Suppose that for every $n \in \mathbb{N}$ there exists a $k \in \mathbb{N}$ such that $f^{2k}(n)=n+k$, and let $k_n$ be the smallest such $k$. Prove that the sequence $k_1,k_2,\ldots $ is unbounded. [i]Proposed by Palmer Mebane, United States[/i]

2019 Korea Winter Program Practice Test, 1

Tags: incenter , algebra , functional equation , geometry , function

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

1992 IMO Longlists, 8

Tags: functional equation , induction , function , algebra

Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation \[f(f(x)) + af(x) = b(a + b)x.\] Prove that there exists a unique solution of this equation.

2019 Swedish Mathematical Competition, 5

Tags: functional , algebra , functional inequality , functional equation , inequalities

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

1990 Swedish Mathematical Competition, 5

Tags: functional , algebra , functional equation

Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2025 Kosovo National Mathematical Olympiad`, P4

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

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 $f(p^{2025})=q^{2025}$.

1989 IMO Longlists, 29

Tags: algebra , functional equation , function , 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 \[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C} \]

2017 Greece Team Selection Test, 3

Tags: algebra , functional equation

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

2004 IMO, 2

Tags: polynomial , functional equation , algebra

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

2024 USA IMO Team Selection Test, 6

Tags: function , functional equation , algebra

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

2010 Contests, 1

Tags: functional equation , functional , algebra