Olympiad problems

2014 Germany Team Selection Test, 2

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

1999 Chile National Olympiad, 7

Tags: functional equation , functional , algebra

Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies: $\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$. $\bullet$ $f (1998) = 2$ Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is satisfied,

2018 China Team Selection Test, 4

Tags: algebra , functional equation , function , fe

Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

1972 IMO Shortlist, 1

Tags: functional equation , function , algebra , functional inequality

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

2016 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

2008 Grigore Moisil Intercounty, 4

Tags: function , functional equation , algebra , find all functions

Given two rational numbers $ a,b, $ find the functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify $$ f(x+a+f(y))=f(x+b)+y, $$ for any rational $ x,y. $ [i]Vasile Pop[/i]

2010 Indonesia TST, 1

Tags: functional equation , functional , algebra

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

2025 Olympic Revenge, 3

Tags: functional equation

Find all $f\colon\mathbf{R}\rightarrow\mathbf{R}$ such that \[f(f(x)f(y)) = f(x + y) + f(xy)\] for all $x,y\in\mathbf{R}$.

OMMC POTM, 2023 2

Tags: algebra , functional equation , function

Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$ $$f(x)f(f(x)+y) = f(x^2) + f(xy).$$ [i]Proposed by Culver Kwan[/i]

2010 Bundeswettbewerb Mathematik, 4

Tags: functional , algebra , polynomial , functional equation , function

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

2020 Taiwan TST Round 1, 2

Tags: function , algebra , functional equation

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds \[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]

1993 IMO Shortlist, 6

Tags: function , functional equation , algebra

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) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

2010 Middle European Mathematical Olympiad, 1

Tags: function , algebra , functional equation

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

2024 ELMO Shortlist, A3

Tags: algebra , functional equation

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$, $$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$ [i]Andrew Carratu[/i]

2019 European Mathematical Cup, 4

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. [i]Proposed by Adrian Beker[/i]

2018 Poland - Second Round, 1

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.

2021 Science ON grade XII, 1

Tags: integration , differentiable function , functional equation , calculus

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

2023 Balkan MO Shortlist, A3

Tags: functional equation , algebra

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

2015 India IMO Training Camp, 2

Tags: functional equation , algebra , function

Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.

2018 Iran Team Selection Test, 1