Olympiad problems

2021 Philippine MO, 3

Denote by $\mathbb{Q}^+$ the set of positive rational numbers. A function $f : \mathbb{Q}^+ \to \mathbb{Q}$ satisfies • $f(p) = 1$ for all primes $p$, and • $f(ab) = af(b) + bf(a)$ for all $ a,b \in \mathbb{Q}^+ $. For which positive integers $n$ does the equation $nf(c) = c$ have at least one solution $c$ in $\mathbb{Q}^+$?

2016 Turkey Team Selection Test, 5

Tags: number theory , function , functional equation , divisibility

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .

2006 Germany Team Selection Test, 1

Tags: functional equation , algebra , function

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

PEN K Problems, 28

Tags: function , functional equation

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

2023 Mongolian Mathematical Olympiad, 1

Tags: functional equation , algebra , cauchy functional equation , three variables

Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\] for all real numbers $x,y,z$.

2004 Cuba MO, 8

Tags: algebra , functional , functional equation

Determine all functions $f : R_+ \to R_+$ such that: a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$ b) $f(2) = 0$ c) $f(x) \ne 0$ for $0 \le x < 2$.

2019 Estonia Team Selection Test, 3

Tags: algebra , functional equation , functional

Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

2016 Azerbaijan Balkan MO TST, 4

Tags: functional equation , algebra , function

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$

2023 SG Originals, Q4

Tags: algebra , functional equation

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

2021 Irish Math Olympiad, 5

Tags: functional equation , algebra , functional

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

2008 Thailand Mathematical Olympiad, 6

Tags: algebra , functional , functional equation

Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$ Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.

2009 Turkey Team Selection Test, 1

Tags: function , algebra , functional equation , algorithm , euclidean algorithm , continued fraction , number theory

Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.

2022 Balkan MO Shortlist, A4

Tags: functional equation , algebra

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2023 Turkey Olympic Revenge, 1

Tags: algebra , functional equation , function , function in r

Find all $c\in \mathbb{R}$ such that there exists a function $f:\mathbb{R}\to \mathbb{R}$ satisfying $$(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)$$ for all $x,y\in \mathbb{R}$. [i]Proposed by Kaan Bilge[/i]

2003 Singapore Team Selection Test, 3

Tags: functional equation , algebra , functional

Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.

1998 IMO, 6

Tags: number theory , functional equation , prime number , algebra

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

2014 Ukraine Team Selection Test, 11

Tags: functional , algebra , functional equation

Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.

2018 USA Team Selection Test, 2

Tags: probability , functional equation , random walk , function , algebra

Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$, \[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\] [i]Proposed by Yang Liu and Michael Kural[/i]

2012 Switzerland - Final Round, 2

Tags: functional equation , functional , algebra

Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

2006 Germany Team Selection Test, 2

Tags: functional equation , algebra

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$. [i]Proposed by B.J. Venkatachala, India[/i]

2012 Balkan MO Shortlist, N3

Tags: function , functional equation , inequalities

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

2017 Morocco TST-, 6

Tags: number theory , polynomial , functional equation , sum of digits

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2012 IMO Shortlist, A1

Tags: functional equation , algebra

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

2004 Austrian-Polish Competition, 7

Tags: algebra , relatively prime , number theory , functional equation

Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers: \[f(x+y) = f(x+1) + f(y+1).\]

2024 Baltic Way, 2

Tags: functional equation , algebra

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[ \frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1 \] for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.