Olympiad problems

2017-IMOC, N1

If $f:\mathbb N\to\mathbb R$ is a function such that $$\prod_{d\mid n}f(d)=2^n$$holds for all $n\in\mathbb N$, show that $f$ sends $\mathbb N$ to $\mathbb N$.

2014 Nordic, 1

Tags: function , algebra , functional equation

Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.

2020 Czech-Austrian-Polish-Slovak Match, 4

Tags: functional , functional equation , algebra

Let $a$ be a given real number. Find all functions $f : R \to R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in R$. (Walther Janous, Austria)

2011 ELMO Shortlist, 2

Tags: function , functional equation , algebra , induction

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

2020 Korea National Olympiad, 1

Tags: algebra , function , functional equation

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$ for all $x,y\in\mathbb{R}$.

2021 Regional Olympiad of Mexico Center Zone, 6

Tags: number theory , functional equation , algebra , sequence

The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: [list] [*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ [*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ [/list] Prove that $a_n=n$ for all positive integers $n$. [i]Proposed by José Alejandro Reyes González[/i]

2013 Dutch BxMO/EGMO TST, 4

Tags: algebra , functional equation , determine all functions

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

2014 Peru IMO TST, 14

Tags: algebra , functional equation , number theory

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 $n$.

2012 Albania Team Selection Test, 5

Tags: algebra , function , functional equation

Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: \[ x,y > 0 \qquad f(x+f(y)) = yf(xy+1). \] a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$. b) Find all such functions that require the given condition.

2023 Indonesia TST, A

Tags: functional equation , function , algebra

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied \[f(x+y) + f(x)f(y) = f(xy) + 1 \] $\forall x, y \in \mathbb{R}$

2009 Tuymaada Olympiad, 4

Tags: polynomial , algebra , floor function , functional equation

Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

1997 Singapore Team Selection Test, 3

Tags: functional equation , algebra , functional

Let $f : R \to R$ be a function from the set $R$ of real numbers to itself. Find all such functions $f$ satisfying the two properties: (a) $f(x + f(y)) = y + f(x)$ for all $x, y \in R$, (b) the set $\{ \frac{f(x)}{x} :x$ is a nonzero real number $\}$ is finite

2024 Brazil Team Selection Test, 4

Tags: algebra , functional equation , function

Find all pairs of positive integers $ (a, b) $ such that $ f(x) = x $ is the only function $ f : \mathbb{R} \to \mathbb{R} $ that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, $ f^n(x) $ represents the function obtained by applying $ f $ $ n $ times to $ x $. That is, $ f^1(x) = f(x) $ and $ f^{n+1}(x) = f(f^n(x))$ for all $n \geq 1$.

2010 Germany Team Selection Test, 3

Tags: algebra , trigonometry , functional equation , function

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

2009 Belarus Team Selection Test, 1

Tags: algebra , functional , functional equation

Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$. I.Voronovich

2006 Switzerland - Final Round, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$

2013 Saudi Arabia BMO TST, 2

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$

2016 Taiwan TST Round 2, 2

Tags: function , algebra , functional equation

Find all function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)$ for all integer $x,y$

1978 IMO, 3

Tags: algebra , functional equation , sequence , partition

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

2012 Germany Team Selection Test, 3

Tags: function , functional equation , algebra

Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

2024-IMOC, N6

Tags: functional equation , number theory

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\] holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) [i]Proposed by supercarry & windleaf1A[/i]

2023 Balkan MO, 1

Tags: functional equation , algebra

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

2021 Thailand TST, 3

Tags: functional equation , nonnegative integers , number theory

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

2024 Canadian Mathematical Olympiad Qualification, 1

Tags: functional equation , algebra

Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$

1994 Swedish Mathematical Competition, 6

Tags: functional , number theory , functional equation

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?