Olympiad problems

2017 Brazil Team Selection Test, 4

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2020 Bulgaria Team Selection Test, 5

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

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 $x \in \mathbb{R}$

2024 USA IMO Team Selection Test, 6

Tags: function , algebra , functional equation

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]

PEN K Problems, 17

Tags: functional equation , function

Find all functions $h: \mathbb{Z}\to \mathbb{Z}$ such that for all $x,y\in \mathbb{Z}$: \[h(x+y)+h(xy)=h(x)h(y)+1.\]

2021 Science ON grade X, 4

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

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]

2005 Austrian-Polish Competition, 6

Tags: functional equation , algebra , cauchy functional equation , sum of powers

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$ \[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]

1968 IMO, 5

Tags: function , algebra , periodic function , functional equation

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2002 Irish Math Olympiad, 3

Tags: function , functional equation , algebra

Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that: $ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.

2025 Euler Olympiad, Round 2, 3

Tags: algebra , functional equation , trigonometry

Find all functions $ f : \mathbb{R} \to \mathbb{R} $ such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

2019-IMOC, A5

Tags: functional equation , positive integer fe , iterated fe , algebra

Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity $$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$ holds for all $x,y \in \mathbb N$

2013 Chile National Olympiad, 4

Tags: functional , functional equation , number theory , algebra

Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$. a) Prove that $f(n)$ is an integer for all $n$. b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.

1988 Bulgaria National Olympiad, Problem 6

Tags: polynomial , fe , functional equation , algebra

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

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 $g(1)=-8$

PEN K Problems, 28

Tags: functional equation , function

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]

2010 Indonesia TST, 2

Tags: function , induction , algebra , functional equation

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]

2017 Canadian Mathematical Olympiad Qualification, 3

Tags: algebra , functional equation , function , difference of squares , special factorizations

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$. $$(x+y)f(x-y) = f(x^2-y^2).$$

2002 IMO, 5

Tags: functional equation , algebra

Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.

2020 Centroamerican and Caribbean Math Olympiad, 3

Tags: algebra , functional equation , function

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

2007 Italy TST, 3

Tags: function , polynomial , functional equation , algebra

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2016 Latvia Baltic Way TST, 4

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

2014 Baltic Way, 4

Tags: functional equation , algebra

Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$

2012 ELMO Shortlist, 8

Tags: function , modular arithmetic , induction , functional equation , algebra

Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

2010 Romania National Olympiad, 4

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

Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that \[f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.\] a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$. b) Prove that $\mathcal{F}$ has exactly two elements. [i]Nelu Chichirim[/i]

2019 Brazil Team Selection Test, 1

Tags: algebra , functional equation

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

2012 Baltic Way, 5

Tags: algebra , functional equation

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which \[f(x + y) = f(x - y) + f(f(1 - xy))\] holds for all real numbers $x$ and $y$.